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February  2019, 13(1): 197-210. doi: 10.3934/ipi.2019011

## Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators

 1 Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong SAR, China 2 Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland 3 Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong SAR, China

Received  May 2018 Revised  October 2018 Published  December 2018

Let $A∈{\rm{Sym}}(n× n)$ be an elliptic 2-tensor. Consider the anisotropic fractional Schrödinger operator $\mathscr{L}_A^s+q$, where $\mathscr{L}_A^s: = (-\nabla·(A(x)\nabla))^s$, $s∈ (0, 1)$ and $q∈ L^∞$. We are concerned with the simultaneous recovery of $q$ and possibly embedded soft or hard obstacles inside $q$ by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain $Ω$ associated with $\mathscr{L}_A^s+q$. It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential $q$. If multiple measurements are allowed, then the surrounding potential $q$ can also be uniquely recovered. These are surprising findings since in the local case, namely $s = 1$, both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are long-standing problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schrödinger operators.

Citation: Xinlin Cao, Yi-Hsuan Lin, Hongyu Liu. Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators. Inverse Problems & Imaging, 2019, 13 (1) : 197-210. doi: 10.3934/ipi.2019011
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